A294783 Number of trees with n bicolored nodes and f nodes of the first color. Triangle T(n,f) read by rows, 0<=f<=n.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 4, 6, 4, 2, 3, 9, 15, 15, 9, 3, 6, 20, 43, 51, 43, 20, 6, 11, 48, 116, 175, 175, 116, 48, 11, 23, 115, 329, 573, 698, 573, 329, 115, 23, 47, 286, 918, 1866, 2626, 2626, 1866, 918, 286, 47, 106, 719, 2609, 5978, 9656, 11241, 9656, 5978, 2609, 719, 106, 235, 1842

Offset: 0

Author

R. J. Mathar, Apr 16 2018

Keywords

• nonn
• tabl

Examples

			The triangle starts
    1;
    1,   1;
    1,   1,   1;
    1,   2,   2,    1;
    2,   4,   6,    4,    2;
    3,   9,  15,   15,    9,    3;
    6,  20,  43,   51,   43,   20,    6;
   11,  48, 116,  175,  175,  116,   48,  11;
   23, 115, 329,  573,  698,  573,  329, 115,  23;
   47, 286, 918, 1866, 2626, 2626, 1866, 918, 286, 47;
  106, 719,2609, 5978, 9656,11241, 9656,5978,2609,719,106;
  235,1842,
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1274

Crossrefs

Cf. A038056 (row sums), A000055 (diagonal and 1st column), A000081 (subdiagonal and 2nd column), A303833 (3rd column), A303843 (4th column), A304311 (connected graphs), A304489 (rooted).

Programs

PARI

R(n, y)={my(v=vector(n)); v[1]=1; for(k=1, n-1, my(p=(1+y)*v[k]); my(q=Vec(prod(j=0, poldegree(p,y), (1/(1-x*y^j) + O(x*x^(n\k)))^polcoeff(p,j)))); v=vector(n, j, v[j] + sum(i=1, (j-1)\k, v[j-i*k] * q[i+1]))); v;}
M(n)={my(B=(1+y)*x*Ser(R(n,y))); 1 + B - (B^2 - substvec(B, [x,y], [x^2,y^2]))/2}
{ my(A=M(10)); for(n=0, #A-1, print(Vecrev(polcoeff(A, n)))) } \\ Andrew Howroyd, May 12 2018

Formula

T(n,f) = T(n,n-f), flipping all node colors.

Extensions

Row 10 completed. - R. J. Mathar, Apr 29 2018